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Metastable molecular hydrogen anionThe lowest states (i.e. the ones with longest life-time) are collected in the table below. The same values were found both from Fano fit and from the projection of the full Green’s function on adiabatic state in the outer well.
Metastable molecular deuterium anionAll the parameters below were calculated from the projection of the full Green’s function on the adiabatic state in the outer well
Two representative examples of cross sections:
• Elastic e- + H2 (J=21, v=2) – boomerang oscillations turn into narrow resonances
• Elastic e- + H2 (J=25, v=1) – only narrow resonances present
For each J we fit the cross section in 100 points in the vicinity of resonances to a Fano profile. Resulting positions and width of resonances are shown together with potentials in right part of this panel.
METASTABLE MOLECULAR HYDROGEN ANIONMartin Čížek, Jiří Horáček, Wolfgang Domcke
Institute of Theoretical Physics, Charles University, Prague, Czech Republic
Institute of Physical and Theoretical Chemistry, Technical University of Munich, Germany
I. MotivationExperimental motivation
• Hurley 1974 – observation of H2- from low-energy arc source.
• Aberth et al. 1975 – observation of HD-, D2- from (> 10s).
• Bae et al. 1984 – existence of D2- not confirmed in two-step experiment
designed to produce metastable quartet state (< 2×10-11s).
• Wang et al. 2003 – observed signature of H2- in signal from discharge plasma.
Theoretical motivation
• H2- is unstable for internuclear separations R close to the equilibrium distance
of H2 (1.4 a0) and decays within few fs. H+H- is stable electronically for R>3a0 but the nuclei can move freely in an attractive polarization force towards smaller internuclear separations R.
• Signatures of narrow resonances are seen in calculated H+H- cross sections for nonzero angular momentum J (see the figures below).
Wave functions indicate the existence of four lower lying resonances with possibly much higher lifetime. Such resonances are known to exist in electron scattering from the HCl molecule for J=0 and were confirmed in experiments of Allan 2000.
GOAL: To find the resonances and their lifetimes for various J and for both H2-, D2
-
e-+H2(J=23) H-+H
II. TheoryBasic equationsNonlocal resonance theory (see Domcke 1991) is employed, with the model described by Čížek et al. 1998. The theory is based on selection of the discrete electronic state φd describing the diabatic transition of the bound H+H- state into the resonance in H2+e- electronic continuum φε . The electronic hamiltonian Hel is then completely described by its components within this basis
The vibrational dynamics is then solved for the projection d of the complete wave function of the system on the discrete state φd
where r stands for all electronic coordinates. The function d is the unique solution of the time-independent Schrödinger (Lippmann-Schwinger) equation with the effective hamiltonian
Nonzero angular momentum J in is taken into account by adding the centrifugal term J(J+1)/2μR both to V0(R) and Vd(R). It is also useful to know the adiabatic potential Vad(R) within the model, given implicitly by
Determination of resonance parametersThe position and the width of a narrow resonance can reliably be obtained from a cross section shape fitting the Fano formula
Many resonances studied here are too narrow to be obtained in this way. Since the resonances can well be understood as metastable states trapped in the outer well, a simple method can be used to obtain the position and width directly. First we calculate nuclear wave functions res(R) and the corresponding energies Eres for the adiabatic bound states in the potential Vad(R). The results for J=23 are collected in the following table
The energies obtained from this procedure compare very well with the ones obtained from the Fano fit to cross sections. Also shown in the table is the estimate of the width from the local complex potential approximation (LCP)
The results are not very good compared to the Fano fit. The Born-Oppenheimer approximation breaks down at small R. It is well known that the LCP can’t describe accurately the dynamics of anions at small R which is the region responsible for the decay of the resonances.
The accurate estimates of Γres for very narrow resonances were obtained from the imaginary part of the projection of the complete Green’s function at energy Eres on the adiabatic state res(R).
),(RVH ddeld .)(0 RVH el),(RVH deld
,),(),()( drRrrRR dd
.)(]0)[()( 10 dRViVTERVRVT dNddN
.)()(
)(..)()(
0
2
dRVRV
RVpvRVRV
ad
ddad
Acknowledgement:I would like to thank prof. Xuefeng Yang for drawing my attention to the problem of long-lived anionic states of hydrogen molecule and to experimental evidence for their existence.
This work was supported by Czech National Grant Agency under project no. GAČR 202/03/D112
Contact personE-mail: [email protected]
WWW: http://utf.mff.cuni.cz/~cizek/
III. VE cross sections and Interpretation
Γ0=2.7×10-4eVΓ0=5.2×10-5eV
Γ1=3.5×10-4eV
Γ2=9.4×10-4eV
Γ3=1.3×10-3eV
Γ4=1.3×10-3eV
Γ0=6.0×10-6eV
Γ1=3.9×10-5eV
Γ2=9.6×10-5eV
Γ3=1.2×10-4eV
Γ4=1.0×10-4eV
Γ0=7.6×10-7eV
Γ1=3.2×10-6eV
Γ2=5.2×10-6eV
Γ3=1.1×10-5eV
Γ0=6.0×10-8eV
Γ1=2.1×10-7eV
Γ2=4.2×10-6eV
Γ0=2.7×10-9eV
Γ1=1.9×10-6eV
Γ0=3.2×10-7eV
Internuclear distance R (a.u.)Po
tent
ial e
nerg
y (e
V)
,1
)( 2
2
baqE
.21
res
resEE
References:
Allan M, Čížek M, Horáček J, Domcke W, 2000, J. Phys. B 33, L209.
Aber W, Schnitzer R, Anbar M, 1975, Phys. Rev. Lett. 34, 1600.
Bae YK, Coggiola MJ, Peterson IR, 1984, Phys. Rev. A 29, 2888.
Čížek M, Horáček J, Domcke W, 1998, J. Phys. B 31, 2571.
Domcke W, 1991, Phys. Rep. 208, 97.
Fano U, 1961, Phys. Rev. 124, 1866.
Hurley RE, 1974, Nuclear Instruments and Methods 118, 307.
Wang W, Belyaev AK, Xu Y, Zhu A, Xiao C, Yang X-F, 2003, Chem. Phys. Lett. 377, 512.
Table I: Parameters of H2- states
J Eres (relative to DA) τ
21 -136 meV 2.4 ps
22 -105 meV 12 ps
23 -75 meV 0.11 ns
24 -47 meV 0.9 ns
25 -20 meV 12 ns
26 5 meV 0.52 μs
27 28 meV 2 ns
Table II: Parameters of D2- states
J Eres(relative to DA) τ
31 -118 eV 0.13 ns
32 -97 eV 0.70 ns
33 -76 eV 6 ns
34 -55 eV 39 ns
35 -35 eV 0.51 μs
36 -16 eV 5.7 μs
37 2 eV 14 μs
38 19 eV 7.2 μs
39 34 eV 41 ps
IV. Summary V. Conclusions• Narrow resonances were found in both VE and DA cross sections with lifetimes by many orders of magnitude larger than for previously known resonances.• The resonances can well be understood as adiabatic states trapped in an outer well separated from the e- + H2 autoionisation region by inner barrier and separated from dissociation into H + H- by an outer centrifugal barrier.• The decay into the e- + H2 channel is controlled by nonlocal dynamics and estimates from adiabatic (local complex) potential give an order of magnitude estimate at best.• The lifetimes of the states reach the values of 0.5 μs and 14 μs for H2
- and D2
- respectively. Even larger values can be expected for T2-.
• Our interpretation of the states explains the lack of a molecular-anion signal in the experiments of Bae et al. 1984.
Open questions - theory• The stability of the states with respect to collisions with other H atoms or H2 molecules is unknown.
• State to state rates for creation/destructions of ions are needed for modeling of equilibrium plasma densities. • Highly rotating anions in other systems ?
Suggestions - experiment• The existence and interpretation of the anions should be confirmed in new experiments – best with measurement of energies and lifetimes. • It is very difficult to create the anions in electron attachment to H2. It is probably much easier to create the states in H- + H2 collisions. The cross sections are unknown, however (to be the subject of further study).
v LCP: Eres Fano: Eres LCP: Γres Fano: Γres
0 -0.075362 -0.075294 1.662×10-5 6.020×10-6
1 -0.037674 -0.037587 9.168×10-5 3.912×10-5
2 -0.011331 -0.011244 2.174×10-4 9.611×10-5
3 0.005578 0.005701 2.861×10-4 1.227×10-4
4 0.015078 0.015055 2.414×10-4 1.007×10-4
.)(2)()(
2)(
0resRVRVdres
LCPres
adRV
Potentials for J=0 Potential Vad(R) for nonzero J